Date of Award
Doctor of Philosophy (PhD)
Katherine E. Stange
This dissertation explores relations between hyperbolic geometry and Diophantine approximation, with an emphasis on continued fractions over the Euclidean imaginary quadratic fields, Q(√−d), d = 1, 2, 3, 7, 11, and explicit examples of badly approximable numbers/vectors with an obvious geometric interpretation.
The first three chapters are mostly expository. Chapter 1 briefly recalls the necessary hyperbolic geometry and a geometric discussion of binary quadratic and Hermitian forms. Chapter 2 briefly recalls the relation between badly approximable systems of linear forms and bounded trajectories in the space of unimodular lattices (the Dani correspondence). Chapter 3 is a survey of continued fractions from the point of view of hyperbolic geometry and homogeneous dynamics. The chapter discusses simple continued fractions, nearest integer continued fractions over the Euclidean imaginary quadratic fields, and includes a summary of A. L. Schmidt’s continued fractions over Q(√−1).
Chapters 4 and 5 contain the bulk of the original research. Chapter 4 discusses a class of dynamical systems on the complex plane associated to polyhedra whose faces are two-colorable (i.e. edge-adjacent faces do not share a color). To any such polyhedron, one can associated a right-angled hyperbolic Coxeter group generated by reflections in the faces of a (combinatorially equivalent) right-angled ideal polyhedron in hyperbolic 3-space. After some generalities, we discuss a simpler system, billiards in the ideal hyperbolic triangle. We then discuss continued fractions over Q(√−1) and Q(√−2) coming from the regular ideal right-angled octahedron and cubeoctahedron.
Chapter 5 gives explicit examples of numbers/vectors in Rr×Cs that are badly approximable over number fields F of signature (r, s) with respect to the diagonal embedding. One should think of these examples as generalizations of real quadratic irrationalities, which we discuss first as our prototype. The examples are the zeros of (totally indefinite anisotropic F-rational) binary quadratic and Hermitian forms (the Hermitian case arises when F is CM). Such forms can be interpreted as compact totally geodesic subspaces in the relevant locally symmetric spaces SL2(OF )\SL2(F⊗R)/SO2(R)r × SU2(C)s. We discuss these examples from a few different angles: simple arguments stemming from Liouville’s theorem on rational approximation to algebraic numbers, arguments using continued fractions (of the sorts considered in chapters 3 and 4) when they are available, and appealing to the Dani correspondence in the general case. Perhaps of special note are examples of badly approximable algebraic numbers and vectors, as noted in 5.10.
Chapter 6 considers approximation in Rn (in the boundary ∂Hn of hyperbolic n−space) over “weakly Euclidean” orders in definite Clifford algebras. This includes a discussion of the relevant background on the “SL2” model of hyperbolic isometries (with coefficients in a Clifford algbra) and a discription of the continued fraction algorithm. Some exploration in the case Z3 ⊆ R3 is included, along with proofs that zeros of anisotropic rational Hermitian forms are “badly approximable,” and that the partial quotients of such zeros are bounded (conditional on increasing convergent denominators).
Chapter 7 considers simultaneous approximation in Rr × Cs as a subset of the boundary of (H2)r × (H3)s over a diagonally embedding number field of signature (r, s). A continued fraction algorithm is proposed for norm-Euclidean number fields, but not even convergence is established. Some exploration and experimentation over the norm-Euclidean field Q(√2) is included.
Finally, chapter 8 includes some miscellaneous results re
Hines, Robert, "Applications of Hyperbolic Geometry to Continued Fractions and Diophantine Approximation" (2019). Mathematics Graduate Theses & Dissertations. 74.